I'm quite sure leaving out "that" in the given sentence is totally fine. I've been taught that you can leave out that, who, and which if the world after "that" is not a verb. If it is a verb, you can also use participle as suggested by parent (but if you use participle too often with long sentences, the sentence will become hard to understand). When using the participle, you should never use that, who, or which.

It should also be noted that leaving out "that" (if grammatical correct) can sometime sound wrong to native speakers, but that varies from accent and region since there is no standard (at least as far as I know).